# Bachelor thesis with Prof. Czakon in 2017

Classical limit of the quantum motion of a particle in a constant magnetic field

In principle, Quantum Mechanics should be reducible to classical mechanics in case of large scale motion. In the case of a harmonic oscillator, one can demonstrate this by observing the time evolution of coherent states. The standard solution for quantum motion in a constant magnetic field is, on the other hand, very different from the classical version. Indeed, the quantum wave function is typically translationally invariant in an arbitrary direction in the plane transverse to the magnetic field and does seem to exhibit axial symmetry, which is well known from the classical circular motion.

The purpose of the thesis is to formulate a quantum wave function for motion in a constant magnetic field which simulates the classical behaviour, similarly to what is done in the case of the harmonic oscillator.

**The student will learn:**

- a deeper meaning of the classical limit of Quantum Mechanics
- how to obtain numerical solutions of the Schroedinger equation

**Requirements:**

- good understanding of Quantum Mechanic

Semi-classical approximation to the decay of a quantum system

A standard approximation when modelling decay is that of constant, time-independent, decay probability. In consequence, it is easy to show that an exponential decay law follows. In reality, a purely exponential decay is inconsistent with Quantum Mechanics. During the decay phase, which can be modelled exponentially, semi-classical methods, i.e. the WKB approximation, are often used to obtain the decay probability, in particular in the case of tunneling. These problems are of particular interest to the question of stability of the quantum ground state of the Universe.

The purpose of this thesis is to numerially model the decay of a simple one-dimensional quantum system with tunneling through a potential barrier and confront the results with the WKB approximation. In a perliminary phase of the thesis, the known proofs concerning the decay law in Quantum Mechanics will be reviewed.

**The student will learn:**

- how tunneling phenomena can be described numerically
- how the principles of quantum mechanics induce modifications in

seemingly obvious laws

**Requirements:**

- good understanding of Quantum Mechanics
- interest in numerical methods

A model of multiple parton emissions

As is well known from electrodynamics, charged particles radiate electromagnetic waves if their trajectory is modified, as is the case in collisions. In particle physics, this radiation field must be described by multiple particle emissions. Such emissions are of great importance both in Quantum Electrodymics and in Quantum Chromodynamics. They are typically modelled with the help of a parton shower.

The purpose of this thesis is to develop a simplified parton shower program to simulate multiple particle emission and answer qualitative questions about the behaviour of cross sections in Quantum Chromodynamics. An ambitious goal is to link the parton shower picture with the classical radiation field.

**The student will learn:**

- what is a parton shower
- how to treat independent particle emissions within a numerical program
- in what sense can a particle ensemble be regarded as a classical field

**Requirements:**

- interest in numerical methods
- basic programming ability